**Plasma engineering math problem**

**Question:** A Langmuir probe of 1 mm diameter and 10 mm long is connected vacuum chamber to measure ion saturation current. Find the plasma density from the following information: electronic charge, e = 1.6×10^{-19} C, Boltzmann constant k= 1.38×10^{-23} J/K, T= 300 K (Assume), and ion saturation current 60.55 µA

**Solution:**

Given that

ion saturation current I_{isat}=60.55μA

=60.55 x 10^{-6}A

The radius of the probe r=1/2mm

=5 x 10^{-4 }m

Length of the probe=0.01 m

The area of the probe is

{\begin{array}{l}A=\pi r^2+2\pi rl\\\;\;\;=3.1416\;\times\;(5\times10^{-4})^2+2\;\times\;3.1416\;\times\;5\;\times10^{-4}\times0.01\\\;\;\;=5.22\;\times\;10^{-5\;}m^2\\\end{array}}

{\begin{array}{l}Density\;\;\eta=\frac{I_{isat}}{0.6\;\times\sqrt{\displaystyle\frac{kT_e}M}}\\\\=\frac{60.55\;\times\;10^{-6}}{0.6\;\times1.6\times10^{-19}\times3.22\times10^{-5}\times\sqrt{\displaystyle\frac{1.38\times10^{-23}\times300}{6.69\times10^{-26}}}}\\\\=7.9\times10^{16}\;M^{-3}\;(ans)\\\end{array}}

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